INSTYTUT FIZYKI
Quantum Fluctuations
in Bose Einstein Condensates
Bartªomiej Ole±
PhD Dissertation
Supervisor: dr hab. Krzysztof Sacha
Kraków, 2009
Contents
I Introduction
5
II Atomic - molecular solitons in the vicinity of a Feshbach
reso-nance
6
1 Model 7
2 Soliton-like solutions 9
2.1 Bright soliton solutions . . . 9
2.2 Two bright solitons and dark soliton solutions . . . 11
2.3 Center of mass oscillations in a trap . . . 13
3 Simulations of an experimental production 14
3.1 Production of soliton-like ground states . . . 14
3.2 Atom losses . . . 16
4 Time evolution in an inhomogeneous magnetic eld 17
III Density uctuations and phase separation in a two component
BEC
21
5 Two component Bose Einstein condensate 22
5.1 Number conserving Bogoliubov theory . . . 23
5.2 Bogoliubov ground state in a particle representation . . . 25
6 Density uctuations close to a phase separation transition 27
6.1 Density measurement . . . 27
6.2 Density uctuations in a nite box . . . 29
7 Bogoliubov vacuum state in a trapped system 34
IV Critical uctuations of an attractive Bose gas in a double-well
potential
37
CONTENTS 2
9 Number conserving Bogoliubov theory 39
9.1 Bogoliubov vacuum for symmetric mean eld solutions . . . 40
9.2 Bogoliubov vacuum for asymmetric states . . . 41
10 Continuous description 44 11 Critical uctuations 48 11.1 Density uctuations . . . 49
11.2 Fluctuations of the order parameter . . . 50
V Second order quantum phase transition of a homogeneous Bose
gas with attractive interactions
52
12 Bogoliubov theory 53 13 Eective Hamiltonian in a continuous description 56 14 Critical uctuations 59VI Phase separation in a two-component condensate in a double
well potential
62
15 Phase transition in a mixture of two condensates 62 15.1 Mean eld critical parameter . . . 6215.2 Bogoliubov ground state for symmetric solutions . . . 63
15.3 Bogoliubov vacuum for phase separated condensates . . . 64
15.4 Exact diagonalization of the two mode model . . . 65
16 Critical uctuations 66 16.1 Density uctuations in the weakly interacting regime . . . 66
16.2 Order parameter and number uctuations . . . 68
17 Entanglement in the double well system 68
VII Self-localization of impurity atoms in a trapped condensate
71
18 Bosonic impurities in a trapped condensate 72 18.1 Phase separation and self-localization of impurity atoms . . . 7219 Signatures of self-localization 78
Part I
Introduction
When describing a many body system forming a condensate a product state is usually assumed which reveals all atoms described by the same single particle wavefunction [1, 2, 3, 4]. The fraction of non-condensed atoms is, however, always present [5]. It can be due to a nonzero tem-perature of the sample, but also at zero temtem-perature interparticle interactions can be responsible for it. We will study only quantum uctuations neglecting all thermal eects.
All chapters except for the rst one discuss systems where quantum uctuations are included in the many body description, and in fact are pronounced. Those systems are rst of all two component condensates where the interplay between the interactions results in reach behaviour [6]. Quantum uctuations occur close to a quantum phase transition [7, 8]. A critical point marking occurrence of mean eld solutions that break symmetry of a trap will be investigated for a condensate with attractive interactions trapped in a one-dimensional symmetric double well potential [9]. Phase transition of a similar origin will be discussed for a condensate in a three dimensional homogeneous case [10] and for a two component system in the double well. In the last chapter self-localized states of impurity atoms immersed in a large condensate are investigated. Only the rst chapter is based purely upon the mean eld approximation, and deals with a Feshbach resonant molecule production [11], ground state properties and their dynamical production.
Each of the chapters begins with an introduction that presents the state of research in the system under study, motivations and main goals. Conclusions from all the chapters are gathered together at the end of the thesis.
6
Part II
Atomic - molecular solitons in the vicinity
of a Feshbach resonance
A set of quantum numbers de ning an internal state of two scattering atoms de nes a scattering channel. If a dissociation threshold of an interatomic interaction potential is at lower energy than the energy of two scattering atoms, the atoms are in an open scattering channel. A Feshbach resonance occurs when the energy of a bound state in a closed channel approaches the energy of the scattering atoms. Due to a coupling between the two channels a resonant formation of the bound state takes place [12, 13]. The coupling can be realized by means of optical transitions (optical Feshbach resonances [14]) or by atomic hyper ne interactions (magnetically tunable Feshbach resonances, see [15]). In the latter, interaction potentials associated with dierent scattering channels can be shifted with respect to one another by means of an external homogeneous magnetic eld, due to the Zeeman eect [16]. It requires a dierence in magnetic moments of a bound state and a pair of free atoms.
A signature of resonant pair formation is divergent behaviour of the scattering length which is a relevant parameter of interactions in cold atom collisions
a(B) = abg µ 1 − ∆B B − Br ¶ , (1)
where abg is an o-resonant value of the scattering length and ∆B is the resonance width.
Note that usually the magnetic eld value where the scattering length has singularity Br is
dierent than B0, magnetic eld for which the scattering energy is degenerate with an energy
of the bound state [16]. The bound state exists for positive values of the scattering length. In experiments Feshbach resonance position is determined from enhanced atom losses, due to bound state formation and inelastic collisions maximal for the divergent scattering length [17] or from the size variation of the trapped cloud [18].
In some cases tuning of the scattering length is necessary either to produce large condensates or to achieve BEC at all [18, 19, 20]. Feshbach resonance techniques are now widely used as a tool to tune interatomic interactions, from an ideal gas [19] to strongly interacting systems with even an attractive nonlinearity leading to a condensate collapse [21, 22]. Control over interaction strengths made it possible to observe long lived Bloch oscillations in an optical lattice potential [23, 24]. Feshbach resonances in ultracold Fermi gases [25] led to observations of BEC of molecules that are remarkably stable against three body decay [26, 27]. In two component Fermi gases a transition region from fermionic super uidity a < 0 to a molecular condensate a > 0 can be studied, and in the so-called BEC-BCS crossover the gas is strongly interacting [28].
In bosonic species coherent oscillations between atom pairs and Feshbach molecules were rst observed in a BEC of 85Rb atoms [29]. Molecules were more directly observed in experiments
using magnetic eld ramps towards a resonance [30] together with, however, signi cant inelastic losses [31].
Based upon the mean eld description of a Bose Einstein condensate many nonlinear phe-nomena can be investigated in BEC, for instance vortices [32, 33, 34], four-wave mixing [35, 36], Josephson-like oscillations [37, 38] or solitons. The latter have been realized experimentally in a quasi one-dimensional condensate with eectively attractive interactions (bright solitons) [39, 40] and with repulsive interactions (dark solitons) [41, 42]. There are studies of solitons in BEC with a time-dependent scattering length [43, 44], in condensates con ned in an eectively two- or three-dimensional trap [45, 46] or in Bose Fermi mixtures [47]. In periodic optical lattice potentials gap solitons were prepared [48].
In the context of nonlinear optics there exists another type of solitonic solutions. It is a parametric soliton which is formed due to a coupling between electromagnetic elds propagating in a nonlinear medium [49] so that the elds propagate as solitons. In the present chapter we will consider parametric solitons occurring as a result of the coupling between atoms and molecules close to a Feshbach resonance.
Analytical solitonic solutions exist in a free one dimensional Feshbach resonance system of noninteracting atoms. We will show that bright, as well as dark soliton solutions can be found numerically in a quasi-1D trapped interacting system. We will simulate production of soliton-like states in a magnetic eld ramp experiment and discuss their subsequent detection [11]. We will qualitatively discuss the problem of atom losses close to the Feshbach resonance.
1 Model
We will use a model commonly applied to the association of atoms into molecules in Bose Einstein condensates, and based upon a description of atoms and molecules in terms of two separate quantum elds [16, 50], ˆψa and ˆψm, respectively,
ˆ H = Z d3r à ˆ ψ†a " −¯h 2 2m∇ 2+ U a(~r) + λa 2 ψˆ † aψˆa # ˆ ψa+ ˆψm† " −¯h 2 4m∇ 2+ U m(~r) + E + λm 2 ψˆ † mψˆm # ˆ ψm + λamψˆa†ψˆaψˆm†ψˆm+ α √ 2 h ˆ ψ† mψˆaψˆa+ ˆψmψˆa†ψˆa† i! . (2)
In ultra-cold collisions real interatomic potentials that are often unknown precisely can be re-placed by a model potential correctly reproducing their crucial properties. We will use a contact pseudopotential parameterized only by the scattering length [51]. Consequently the strength of the atomic, molecular and atomic-molecular interaction is given by the coupling constants proportional to respective scattering lengths
λa= 4π¯h2a/m, λm = 2π¯h2am/m, λam = 4π¯h2aam/mr. (3)
Here m and mrstand for the atomic and reduced mass, respectively. The scattering lengths a, am,
1 MODEL 8
far from resonances [16]. Since precise values of the molecular and atomic - molecular scattering lengths are known only for very few resonances, we will assume λm = λam = λa. The trapping
potentials for atoms and molecules are Ua(~r) and Um(~r), respectively. The Hamiltonian (2)
contains also a term describing the association and decay of molecules, whose strength depends on the resonance width ∆B, dierence between molecular and atomic magnetic moments, ∆˜µ = ˜
µm− 2˜µa and the background scattering length a,
α = s
4π¯h2a∆˜µ∆B
m . (4)
In case of a magnetically tunable Feshbach resonance, the magnetic eld dependence of the Hamiltonian comes only from the detuning E, the dierence between a bound state energy and the energy of a free atom pair. Note that the magnetic eld B0 corresponding to ε = 0 is in
general shifted with respect to the magnetic eld Br which reveals the divergent value of the
scattering length [16].
In the following we will focus on a Feshbach resonance observed in a87Rbcondensate at the
magnetic eld Br = 685.43 G. The resonance width is ∆B = 0.017 G, the background scattering
length a = 5.7 nm and ∆˜µ = 1.4µB [52], where µB is the Bohr magneton. Atoms are prepared
in an |f, mfi = |1, 1i entrance channel which is the hyper ne ground state of 87Rb.
We assume a quasi one-dimensional trap, i.e. the transverse con nement so strong that transverse dynamics is reduced to the lowest state in the trap. We take harmonic trapping potentials with frequencies ωm,⊥ = ωa,⊥ = 2π × 1500 Hz and ωm,x = ωa,x = 2π × 10 Hz 1.
With the chosen trap parameters the system is eectively one-dimensional. Throughout the chapter we will apply the mean eld approximation replacing the quantum elds in their equations of motion by their expectation values φa =
D ˆ ψa E , φm = D ˆ ψm E . In the units E0 = ¯hωa,x, x0 = q ¯h/mωa,x, τ0 = 1/ωa,x, (5)
the dimensionless stationary mean eld equations are the following µφa = " −1 2 ∂2 ∂x2 + 1 2x 2+ λ aNφ2a+ λamNφ2m # φa+ α √ 2Nφmφa 2µφm = " −1 4 ∂2 ∂x2 + x 2+ ε + λ mNφ2m+ λamNφ2a # φm+ α s N 2φ 2 a, (6)
where µ is the chemical potential of the system that stems from the conservation of the total number of atoms N in the system. Searching only for ground states we have restricted to real
1The same trap frequencies for atoms and molecules are chosen for simplicity. Dierent frequencies in the
transverse directions would lead only to modi cation of the eective coupling constants in the 1D equations (6). We will see that dierent frequencies in the longitudinal direction do not introduce any noticeable changes in shapes of solitons as long as the widths of the soliton wavepackets are much smaller than the characteristic length of the traps.
wavefunctions. The magnetic eld detuning in the one-dimensional situation is modi ed by the trap frequencies ε = E + (ωa,⊥− 2ωm,⊥)/ωa,x = E − 150. The wave-functions φa(x) and φm(x)
are normalized so that
Z
dx³|φa(x)|2+ 2|φm(x)|2
´
= 1. (7)
Equations (6) constitute a Feshbach resonance version of the GrossPitaevskii equation [1, 2, 3, 4], commonly used for describing almost perfect condensates, i.e. condensates with negligible depletion eects [5]. The model (6) neglects also eects of particle losses due inelastic collisions. They could be introduced via imaginary loss coe cients [53], provided there is an experimental analysis of losses in the vicinity of the Feshbach resonance of interest. The dimensionless 1D cou-pling constants can be calculated integrating out transverse wavefunctions which were assumed to be ground states of the trapping potentials. For the present choice of the system parameters they have the values
λa= λm = λam ≈ 0.505, α ≈ 41.0. (8)
The dynamics of the system we will study using time dependent mean eld equations i∂φa ∂t = " −1 2 ∂2 ∂x2 + 1 2x 2+ λ aN|φa|2+ λamN|φm|2 # φa+ α √ 2Nφmφ∗a i∂φm ∂t = " −1 4 ∂2 ∂x2 + x 2 + ε + λ mN|φm|2+ λamN|φa|2 # φm+ α s N 2 φ 2 a. (9)
2 Soliton-like solutions
A soliton is a special solution to a nonlinear equation, which preserves its shape during time evolution and reveals elastic scattering from another function of its type. Analytic solitonic solutions can be found only for a special choice of the system parameters. Wavefunctions found in other cases numerically, that do not spread in time evolution, we will call soliton-like solutions. We will not study soliton collisions. One can expect that their scattering properties will depend on an amount of the energy involved in a collision, i.e. whether it is high enough to produce also transverse excitations.
2.1 Bright soliton solutions
In the absence of trapping potentials and elastic interparticle interactions the set of equations (6) has an analytical solitonic solution
φa(x) = ± A cosh2³x l ´, φm(x) = − A cosh2³x l ´, (10)
2 SOLITON-LIKE SOLUTIONS 10 -0.5 -0.25 0 0.25 0.5 x -3 -2 -1 0 1 2 3 wavefunction (a) (a) -0.5 -0.25 0 0.25 0.5 x -4 -2 0 2 4 wavefunction (a) (b)
Figure 1: Ground states of the coupled system for N = 100 (a) and N = 1000 (b). In both panels φa are marked in black and φm in red. Ground states of the system (6) found numerically
for ε = −900.53 (a) and ε = −1505.25 (b) are drawn with solid lines. Analytic solutions (10) are marked with dashed and variational functions (13) with dotted lines. For N = 100 the gaussian approximation is indistinguishable from the exact eigenstate.
where A = √ 3 2Nαl2, l = µ 18 α2N ¶1/3 , (11)
and the chemical potential µ = −2/l2. It reveals an equal number of atoms and molecules in the
system and is valid for ε = −6/l2. Solitons (10) have been investigated in nonlinear optics [54]
where a classi cation of solitonic solutions to a set of equations describing a parametric waveguide was provided [55], and it was shown that the same model can describe coupled atomic-molecular condensates [56]. Solutions (10) exist also for a problem of an impurity self-localization in a BEC [57].
Time propagating parametric solitons of width l are given by a solution to the time dependent equations (9) φa(x, t) = ± Aeivxe−i(v2+µ)t cosh2³x−vt l ´ , φm(x, t) = − Ae2ivxe−2i(v2+µ)t cosh2³x−vt l ´ , (12)
where v is a propagation velocity.
We have solved the set of equations (6) and (9) numerically and found soliton-like solutions for a broad range of detuning values ε, in general having unequal atom and molecule numbers. Moreover, soliton-like solutions exist also in an interacting system in a trap. The repulsive interactions given by positive values of λa, λm and λam compete with the eectively attracting
transfer term. For small particle numbers N = 100, 1000 the ground states are hardly in uenced by the trap. For particle numbers of the order of N = 10000 and higher the shape of a ground state is determined mainly by the interplay between the repulsive interactions and the trapping potential, and is modi ed by the transfer term. Neglecting the kinetic energy would correspond to the Thomas-Fermi approximation [4].
We have also performed a Gaussian variational analysis [45] taking into account the values (8) and minimizing the system energy for states of the form
φa(x) = ˜Ae−˜ax
2
, φm(x) = ˜Me− ˜mx
2
. (13)
Requiring the same fraction of atoms and molecules, for N = 100 (N = 1000), we obtain ˜
A = 1.77, ˜a = 139.54, ˜M = 1.70, ˜m = 117.43, ε = −900 ( ˜A = 1.08, ˜a = 19.58, ˜M = 0.97, ˜
m = 12.47, ε = −1429).
In Fig. 1 we compare exact ground states of the equations (6) found for coupling constants (8) with the solitonic solutions (10) and the variational functions (13). We can see that for a small particle number (N = 100) it is the attractive transfer term that dominates, for moderate N = 1000the solitonic approximation is too crude, but the gaussian functions work reasonably well in both cases. Note that the widths of the states are much smaller than the harmonic oscillator ground state width, which indicates that even for a very small particle number the nonlinearities in Eqs. (6) determine the shapes of the states. We have veri ed the solitonic character of exact solutions performing time evolution in the presence of the transversal trap and with the axial one turned o.
We have taken λm = λa = λam for simplicity. The studies of an optical resonance in 87Rb
atoms [58] indicate that the atomic-molecular elastic interactions strongly depend on the internal state of a molecule and can even be attractive (λam < 0). Negativity of the latter coupling
constant should in fact make the solitonic behaviour even more pronounced. Gaussian variational calculation based upon (13) con rms the existence of soliton-like states for λm ∈ (0, 2λa) and
λam ∈ (−2λa, 2λa) (for N = 1000), which shows that the choice of equal coupling constants is
not essential in order to deal with soliton-like solutions.
All stationary states of the Gross-Pitaevskii equations (6) have been obtained numerically by means of the imaginary time evolution method, which implies that the obtained ground states are dynamically stable.
2.2 Two bright solitons and dark soliton solutions
In the absence of trapping potentials and interparticle interactions an asymptotic double soliton solution can be found
φa(x) = A cosh2³x−ql ´ ± A cosh2³x+ql ´ φm(x) = − A cosh2³x−ql ´ − A cosh2³x+ql ´, (14)
2 SOLITON-LIKE SOLUTIONS 12 -3 -2 -1 0 1 2 3 x -1 -0.5 0 0.5 1 wavefunction (a) -3 -2 -1 0 1 2 3 x -0.4 -0.2 0 0.2 0.4 wavefunction (b)
Figure 2: Double bright soliton solutions in a trapped interacting system for N = 100 (a) and
N = 1000 (b), ε = 0. In both panels φa are represented by black solid lines and φm by red
dashed ones.
valid for q À l, where
l = µ 36 α2N ¶1/3 , A = √ 3 2Nαl2, (15)
and µ = −2/l2, ε = −6/l2. Due to symmetry of Eqs. (6) the atomic wavefunctions can be even
or odd, whereas the molecular ones must be even (because of the φ2
a term).
Using the rst excited state of the harmonic potential as an initial state for imaginary time evolution makes it possible to numerically nd stationary states of the form (14). We have found double solitonic states also in the interacting system (8) in the trap, and for unequal atom and molecule numbers, see Fig. 2. Note that the widths of the states in panel (a) are much smaller than the rst excited state width of the harmonic oscillator, which indicates that also here it is the nonlinearities that determine the shapes of the states.
With an increasing number of particles the healing length de ned as
ξ = q 1
λaN (|φa|2+ |φm|2)
, (16)
becomes much smaller than the spatial extent of the two-solitonic state. This, in addition to the phase ip along the atomic wavefunction makes density pro les resemble those of dark solitons known for condensates with repulsive interactions [41, 42]. Applying in (6) the Thomas Fermi approximation and using dark soliton pro les, an exact eigenstate can be approximated by the wavefunctions φa(x) ≈ φTFa (x) tanh à x ξ ! ,
-20 -10 0 10 20 x -0.2 -0.1 0 0.1 0.2 wavefunction -0.2 0 0.2 -0.2 0 0.2 (a) -20 -10 0 10 20 x -0.1 -0.05 0 0.05 0.1 wavefunction -0.2 0 0.2 -0.1 0 0.1 (b)
Figure 3: Dark soliton approximation (dashed lines) to eigenstates of (6) (solid lines) for N = 105.
Panel (a) shows the atomic whereas panel (b) the molecular wavefunctions. The insets show a central region of the order of the healing length.
φm(x) ≈ φTFm (x) ¯ ¯ ¯ ¯ ¯tanh à x ξ !¯ ¯ ¯ ¯ ¯, (17) where φTF m (x) = γ 2 − s γ2 4 + 2µ − x2 6λaN , φTF a (x) = v u u t2µ − x2 2λaN − √ 2α λa √ Nφ TF m (x) − [φTFm (x)]2, γ = √ 2 3α√N à x2− 2µ 2 − α2 λa ! . (18)
The corresponding chemical potential can be found from the normalization condition (7) (see Figure 3).
2.3 Center of mass oscillations in a trap
In the case of a harmonic trap (with equal trapping frequencies for atoms and molecules), having any solutions φa0(x) and φm0(x) of the time-independent problem (6), corresponding to the
chemical potential µ, one can nd harmonic oscillations of the center of mass of the particle cloud described by translated functions φa0(x − q) and φm0(x − q). The time evolution is given
by
3 SIMULATIONS OF AN EXPERIMENTAL PRODUCTION 14
φm(x, t) = φm0(x − q)e−i2µte2i[ ˙qx−S(q)], (19)
where S(q) = 1 2 Z t t0 dt0h˙q2(t0) − q2(t0)i, (20) and d2q dt2 + q = 0. (21)
The proof can be done by direct substitution of (19) into (9).
This indicates that, similarly as in the case of the Gross-Pitaevskii equation for a single condensate in a harmonic trap, time evolution of the translated stationary solutions reveals harmonic oscillations of the center of mass of the particle cloud.
3 Simulations of an experimental production
Molecule observation in magnetically tunable Feshbach resonances can be achieved using two experimental schemes. If we start with a pure atomic condensate, a rapid change of the magnetic eld towards its resonant value and back produces a quantum superposition state of atoms and molecules, i.e. an atom - molecule coherent state. Consequently a number of atoms in the atomic fraction oscillates with a frequency corresponding to the molecular binding energy [29, 50, 59]. In the other technique the magnetic eld is slowly varied in the direction from negative to positive scattering lengths so that an initial state adiabatically follows ground states close to a resonance [30, 31, 50, 60]. For the Feshbach resonance under study this requires a magnetic eld to be rapidly changed to an above resonance value (where a < 0) before an actual molecule production.
3.1 Production of soliton-like ground states
We assume that the condensate has already been prepared in its ground state far above the resonance, i.e. B À B0, where the amount of molecules is very small. The timescale of a molecule
production has to be carefully chosen. It should be long enough to preserve the adiabaticity so that we would end up in a solitonic ground state, but also short enough in order not to make loss processes dominant.
We consider a 87Rb condensate around the Feshbach resonance at B
r = 685.43 G and of
the width ∆B = 0.017 G. We have studied production of soliton-like states for N in the range between 100 and 10000. As initial states we took ground states of the model (6) for ε = 3331 and performed time integration of equations (9) assuming dierent ramp speeds of linear magnetic eld changes. The nal detuning value was εend = 0 (so the total change in the magnetic eld
-0.5 -0.25 0 0.25 0.5 x 0 1 2 3 4 5 6 density (a) -0.5 -0.25 0 0.25 0.5 x 0 0.5 1 1.5 2 density (b)
Figure 4: Density pro les of solitons produced after 95.5 ms of the magnetic eld sweeping for
N = 100 (a) and after 93.8 ms for N = 1000 (b) solid lines. Dashed lines show exact ground states corresponding to the magnetic eld at the end of the sweeping. In each panel, the black curves represent atoms and the red ones molecules. The norm in the time evolution is preserved on the level of 10−5. The simulated state contains tiny excitations outside the center, not shown
in the Figure.
eld o the resonance center (i.e. Bf inal > Br) and the fact that ground states for ε = 0 already
reveal soliton-like behaviour. The fraction of atoms in the initial states, R |φa|2, was 99% and
93% for N = 100 and N = 10000, respectively.
The square overlap between nal states from the time evolution and the corresponding ground states in the trap depends nonlinearly on the evolution time. We have found that it approaches unity for evolution times longer than 90 ms, which should be an experimentally ac-cessible timescale [31]. Figure 4 shows simulated soliton-like states that reveal square overlaps 0.94and 0.92 with exact ground states for the nal detuning value, for N = 100, tevol = 95.5ms
and N = 1000, tevol = 93.8 ms, respectively. For N = 10000 the shortest evolution time that
results in a reasonably high squared overlap (i.e. 0.83) is 90.7 ms.
Evolution times shorter than the ones mentioned above might also prove useful. Final states after such nonadiabatic evolution reveal a soliton train structure similar to soliton trains observed in an attractive7Li[40]. Figure 5 shows a state produced during 25.5 ms in a system of N = 1000
atoms. Actually stationary multi-peak soliton solutions should exist for an arbitrary number of peaks. Unfortunately it is not possible to obtain them by means of the imaginary time evolution method. The symmetry properties of the set (6) make an initial state collapse to the ground or the rst excited state having the number of peaks equal to 1 and 2, respectively. The state in Fig. 5 contains one or several excited soliton-like states in the 1D trapping potential.
Solitons created in a quasi-1D trap can be experimentally veri ed when an axial trap is turned o and the measured wavepackets propagate without spreading [39, 40]. We have con rmed that
3 SIMULATIONS OF AN EXPERIMENTAL PRODUCTION 16 -4 -2 0 2 4 x 0 0.5 1 1.5 density
Figure 5: Soliton train obtained for N = 1000 after 25.5 ms magnetic eld sweeping. Solid black line represents the atomic while the dashed red one the molecular density.
soliton-like states produced in the simulations (such as those presented in Fig. 4) preserve their shapes without axial trapping within the time 100 ms that was chosen in the simulations.
3.2 Atom losses
One of the signatures of a Feshbach resonance is an enhanced loss rate of atoms when approaching a resonance value [17]. Atom loss processes can be due to two or three body collisions. Two body inelastic collisions which are signi cant in a 85Rb Feshbach resonance [61] are absent in
Feshbach resonances of 87Rb atoms in the |1, 1i ground state, which is the case here. Three
atoms can scatter to form a molecule and an atom carrying away the excess energy. Both can escape from the trap if their kinetic energy is high enough [62]. Molecules produced by means of a Feshbach resonance are often produced in high vibrational states, which makes them prone to deexcitation or dissociation to noncondensed atoms and possibly a trap escape [63, 64].
Experimental analysis of atom losses is often not straightforward since usually only atoms can be detected. Conversion of atoms into molecules at the broadest resonance in87Rb, subsequent
separation of the two species by means of a Stern-Gerlach eld (i.e. a magnetic eld gradient) and conversion back, reveals a 63% atom loss due to inelastic collisions [31]. In the systematic studies [52] the fraction of atoms lost during a 50 ms hold time at the center of Br = 685.43 G
resonance is 78%. There, however, the detection scheme is dierent. After the 50 ms hold time the magnetic eld is rapidly switched o, the condensate is released from a trap and after another 14ms the atomic fraction is detected. Most probably not all molecules formed in the experiment are dissociated back to atoms, therefore the estimated loss rate is an upper limit to an actual inelastic loss rate. We estimate central densities of particles at resonance in [52] and in our simulations to be of the same order of magnitude, 1015 cm−3. This is because on one hand we
have taken much smaller particle numbers (maximum of 104 compared to N = 4 × 106 in [52]),
but the strong transverse con nement necessary to achieve solitons increases the density with respect to the 3D situation in [52]. A possible way to reduce the peak density could be to reduce the transverse con nement as much as possible. We have checked that even in a transverse trap with a frequency 2π × 200 Hz the condition for a quasi-1D con guration is ful lled. The crucial idea of how to reduce the losses is to nish the time evolution o-resonance, at a magnetic eld already supporting soliton-like states.
Actually, if atoms escaping from a trap do not excite the trapped mixture, solitons can exist even in the presence of losses. This is because the attractive solitonic coupling comes from the transfer term scaling as√N whereas interactions scale as N with the total particle number. The focusing eect of the former should in fact be enhanced by losses.
In conclusion to this section, inelastic losses although signi cant close to the Feshbach res-onance that we are studying, should not prohibit the opportunity of observing atom-molecule solitons. More experimental data and precise theoretical models would be necessary to study the losses near the Feshbach resonance that we have chosen.
4 Time evolution in an inhomogeneous magnetic eld
As already mentioned in the previous section, a magnetic eld gradient is often used to separate atoms and molecules, such as it was done in a 3D setup [31]. We will show that the separation eect resulting from dierent magnetic moment values can be suppressed by the strong coupling in (6) that is responsible for the solitonic states. In the present section we will assume a more general version of the time evolution equations (9). The magnetic eld is now given by two parameters, B, that tells us how far from the resonance we are, and Bgrad which is responsible
for the magnetic eld gradient. Using the units (5) we have i∂φa ∂t = " −1 2 ∂2 ∂x2 + 1 2x 2+ µ˜a E0 (Bgradx0x + B) + λaN|φa|2+ λamN|φm|2 # φa+ α √ 2Nφmφ∗a i∂φm ∂t = " −1 4 ∂2 ∂x2 + x 2 +µ˜m E0 (Bgradx0x + B) + λmN|φm|2+ λamN|φa|2 # φm+ α s N 2φ 2 a. (22)
Note that equations (9) correspond to the situation where Bgrad= 0. It is possible to transform
these equations so that the magnetic eld is present only in one of them. First, a coordinate transformation x → x + βa can be applied. Then, phases of the wavefunctions are adjusted
according to φa→ φaexp µi 2 µ βa2− µ˜aB E0 ¶ t ¶ , φm → φmexp µ i µ βa2− µ˜aB E0 ¶ t ¶ , (23)
where βa = x0µ˜aBgrad/E0, and the equations (22) take nally the form
i∂φa ∂t = " −1 2 ∂2 ∂x2 + 1 2x 2+ λ aN|φa|2 + λamN|φm|2 # φa+ α √ 2Nφmφ∗a
4 TIME EVOLUTION IN AN INHOMOGENEOUS MAGNETIC FIELD 18 (a) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 x 0 0.2 0.4 0.6 0.8 1 t 0 1 2 3 4 5 6 density (a) (b) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 x 0 0.2 0.4 0.6 0.8 1 t 0 0.4 0.8 1.2 1.6 2 density
Figure 6: Time evolution of an initial soliton-like state for N = 100 in a magnetic eld gradient
Bgrad = 1 G/cm, for the detuning εgrad = 2216 (Eq. (26)). Panel (a) shows the atomic
wave-packet, whereas panel (b) the molecular one. The time t = 1 corresponds to 15.9 ms.
i∂φm ∂t = " −1 4 ∂2 ∂x2 + x 2+ βx + ε grad+ λmN|φm|2+ λamN|φa|2 # φm+ α s N 2φ 2 a, (24) where β = x0∆˜µ E0 Bgrad, (25)
and the parameter
εgrad= ε − µ x0 E0 Bgrad ¶2 ˜ µa∆˜µ, (26)
can be regarded as an eective detuning, modi ed with respect to the case without gradient (given by ε). As initial states for the time evolution without the axial trapping but with the transverse traps present we have taken ground states of the system (6) for ε = 0 and particle numbers N = 100 and N = 1000. As we can see from (26) a sudden turn-on of a 1 G/cm magnetic eld gradient modi es the detuning to εgrad = 2216. We have studied the time evolution with
that value, but checked also what happens for a magnetic eld such that εgrad= 0.
In the former case molecules are generally converted back to atoms, see Fig. 6 and 7, which is not surprising because of the positive value of the eective detuning. On the evolution timescale tevol = 15.9 ms, almost all of the initial N = 100 particles end up as individual atoms, the
attractive coupling provided by the atom-molecule transfer term practically disappears, and consequently the atomic wavefunction φa begins to spread. For N = 1000 during the same
evolution time the coupling term is eectively stronger and its competition against the gradient results in the wavefunction splitting and soliton train production (see Fig. 7). Similar splitting
4 TIME EVOLUTION IN AN INHOMOGENEOUS MAGNETIC FIELD 19 (a) -10 -8 -6 -4 -2 0 2 4 6 x 0 0.2 0.4 0.6 0.8 1 t 0 0.4 0.8 1.2 1.6 2 density (b) -10 -8 -6 -4 -2 0 2 4 6 x 0 0.2 0.4 0.6 0.8 1 t 0 0.1 0.2 0.3 0.4 0.5 0.6 density
Figure 7: Time evolution of an initial soliton-like state for N = 1000 in an inhomogeneous magnetic eld Bgrad = 1 G/cm, for the detuning εgrad = 2216 (see Eq. (26)). Panel (a) shows
the atomic wave-packet, whereas panel (b) the molecular one. The time t = 1 corresponds to 15.9 ms. (a) -8 -7 -6 -5 -4 -3 -2 -1 0 1 x 0 0.2 0.4 0.6 0.8 1 t 0 0.2 0.4 0.6 0.8 1 1.2 1.4 density (a) (b) -8 -7 -6 -5 -4 -3 -2 -1 0 1 x 0 0.2 0.4 0.6 0.8 1 t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 density
Figure 8: Time evolution of an initial soliton-like state for N = 1000 in a magnetic eld gradient
Bgrad = 1 G/cm, for the detuning εgrad = 0 (see Eq. (26)). Panel (a) shows the atomic
4 TIME EVOLUTION IN AN INHOMOGENEOUS MAGNETIC FIELD 20
of solitonic wavepackets has been analyzed in the case of a single component BEC in the presence of the gravitational eld [65].
In the case with εgrad = 0 we are eectively on the molecular side of the resonance, the
transfer (coupling) term dominates over the splitting eect caused by the gradient, and as can be seen in Figure 8, the soliton-like state propagates without loosing its shape or splitting to smaller wavepackets. A closer look on the Figure 8 reveals a slow conversion of atoms into molecules.
For a higher eld gradient than the value chosen in the simulations shown in gures 6-8 or for slightly dierent values of the elastic interaction coupling constants λa, λm, λam, we have
observed qualitatively similar behaviour, i.e. no separation to atomic and molecular clouds, as well as the relevance of the εgrad value. We attribute this behaviour to the strongly con ned
Part III
Density uctuations and phase separation
in a two component BEC
Soon after rst experiments that produced a Bose Einstein condensate [66, 67, 68] atoms in two dierent hyper ne states were simultaneously trapped in a magnetooptical trap and subsequent cooling produced a double condensate [69]. Application of optical traps made it possible to investigate spinor condensates where spin changing collisions do not lead to atom escape from a trap [70]. With the additional degree of freedom spin domains could be observed [70, 71] and various magnetic phases in optical lattice potentials were studied theoretically [72]. Collective oscillatory modes were studied in a condensate composed of two dierent atomic species produced by means of sympathetic cooling [73]. Quantum phase transitions in Bose-Fermi mixtures were observed [74, 75].
In a double condensate system atom interferometry experiments were performed [76] where dynamics of the relative phase was investigated [76, 77]. It was shown that two condensates can repel each other and spatially separate, one forming a shell around the other [78]. The binary atomic condensate systems and phase separation were discussed within the mean eld theory [79, 80, 81] and dynamical instabilities leading to phase separation were identi ed from linearized equations of motion [82]. Phase separated con gurations breaking the symmetry of a trap were also found [83, 84, 85, 86].
Ideal condensates are su ciently well described within the mean eld theory and a corre-sponding Gross-Pitaevskii equation [4]. The Bogoliubov theory [87] is usually applied (i) to test for the dynamical stability of mean eld solutions; (ii) to nd collective excitations such as dipole oscillations in a trap [4]; (iii) to study a quasiparticle excitation spectrum; (iv) and nally to check the initial assumption of all particles occupying the same single particle mode.
In the Bogoliubov theory small quantum corrections to a mean eld solution are introduced. A key idea of the original Bogoliubov theory [87] is a U(1) symmetry breaking assumption that an atomic eld operator has a nonzero expectation value. Such a coherent state involves super-position of states with dierent numbers of atoms, which is in principle far from the experimental reality. Moreover, a careful analysis shows that it involves an eigenvalue problem of an operator which is not diagonalizable. Consequently the theory must break down in nite time [51, 88].
A number conserving version of the Bogoliubov theory overcomes the above problems [89, 90, 91]. It should give the same results for large particle numbers. There are, however, examples where the N-conserving theory works in a regime of the standard approach breakdown [92].
Generalization of the Bogoliubov theory to homogeneous double condensate systems was done in [93, 94] and for the number conserving version in [95].
In the present chapter we will study a homogeneous two component condensate using the number conserving Bogoliubov theory. We will derive the system ground state in the particle
5 TWO COMPONENT BOSE EINSTEIN CONDENSATE 22
representation in the presence of intra- and intercomponent interactions. Based upon the Bo-goliubov vacuum state we will show that in a nite system the phase separation is followed by signi cant long wavelength density uctuations.
5 Two component Bose Einstein condensate
We consider a two component Bose-Einstein condensate formed by a mixture of two kinds of atoms (or the same atoms in two dierent internal states), i.e. Na atoms of type a and Nb atoms
of type b [96]. The Hamiltonian of the system reads ˆ H = Z d3r à ˆ ψa† " − ¯h 2 2ma ∇2+ Va(~r) + ga 2ψˆ † aψˆa # ˆ ψa + ˆψ†b " − ¯h 2 2mb ∇2+ Vb(~r) + gb 2ψˆ † bψˆb # ˆ ψb +g ˆψ† aψˆb†ψˆaψˆb ´ , (27)
where ma, mb are the particle masses, Va(~r), Vb(~r) stand for trapping potentials and
ga = 4π¯h 2a a ma , gb = 4π¯h 2a b mb , g = 2π¯h2aab µ 1 ma + 1 mb ¶ , (28)
where aa, ab, aab are the scattering lengths. Similarly as in the previous chapter we assume that
interparticle interactions are given by the contact potential [51].
The number conserving Bogoliubov theory [51, 89, 95] assumes the following decomposition of the bosonic eld operators
ˆ
ψa(~r) = φa0(~r)ˆa0+ δ ˆψa(~r), ψˆb(~r) = φb0(~r)ˆb0+ δ ˆψb(~r), (29)
where we separate the operators ˆa0 and ˆb0 that annihilate atoms in modes φa0 and φb0,
respec-tively, from quantum corrections δ ˆψa(~r), δ ˆψb(~r)that are assumed small. Consequently the modes
φa0 and φb0 are macroscopically occupied by atoms
hˆa†0ˆa0i ≈ Na, hˆb†0ˆb0i ≈ Nb. (30)
The perturbation expansion of the Hamiltonian in powers of δ ˆψa and δ ˆψb [95] leads to the
following results. Minimizing the system energy in the zero order we obtain coupled Gross-Pitaevskii equations [96] Ha GPφa0 = 0, HGPb φb0= 0, (31) where Ha GP = − ¯h2 2ma ∇2+ V a+ gaNa|φa0|2+ gNb|φb0|2− µa,
HGPb = − ¯h
2
2mb
∇2 + Vb + gbNb|φb0|2+ gNb|φb0|2− µb. (32)
It is possible then to nd the single particle modes macroscopically occupied by atoms together with the corresponding chemical potentials µa and µb. The rst order terms disappear and the
second order terms of the expansion, the actual Bogoliubov theory, will be discussed in the following section.
5.1 Number conserving Bogoliubov theory
Following [89, 95] we can write an eective second order Hamiltonian as ˆ Heff ≈ 1 2 Z d3r³Λˆ† a, −ˆΛa, ˆΛ†b, −ˆΛb ´ L ˆ Λa ˆ Λ† a ˆ Λb ˆ Λ†b , (33) where L = Ha GP+ gaNaQˆa|φa0|2Qˆa gaNaQˆaφ2a0Qˆ∗a g √ NaNbQaφa0φ∗b0Qb g √ NaNbQaφa0φb0Q∗b −gaNaQˆ∗aφ∗2a0Qˆa −HaGP− gaNaQˆ∗a|φa0|2Qˆ∗a −g √ NaNbQ∗aφ∗a0φ∗b0Qb −g √ NaNbQ∗aφ∗a0φb0Q∗b g√NaNbQbφ∗a0φb0Qa g√NaNbQbφa0φb0Q∗a HGPb + gbNbQˆb|φb0|2Qˆb gbNbQˆbφ2b0Qˆ∗b −g√NaNbQ∗bφ∗a0φ∗b0Qa −g √ NaNbQ∗bφa0φ∗b0Q∗a −gbNbQˆ∗bφ∗2b0Qˆb −HGPb − gbNbQˆ∗b|φb0|2Qˆ∗b , (34) and ˆ Qa= 1 − |φa0ihφa0|, Qˆb = 1 − |φb0ihφb0| (35)
are projection operators to non-condensate subspaces. Diagonalization of the eective Hamil-tonian (33) amounts to solving an eigenproblem of the non-hermitian operator L, so-called Bogoliubov-de Gennes equations [51]. The single particle excitation operators creating an atom out of a condensate mode are given by
ˆ Λ† a(~r) = ˆa0 √ Na δ ˆψ† a(~r), Λˆ†b(~r) = ˆb0 √ Nb δ ˆψb†(~r), (36)
and ful l the following commutation relations
[ˆΛa(~r), ˆΛ†a(~r0)] ≈ h~r| ˆQa|~r0i, [ˆΛb(~r), ˆΛ†b(~r0)] ≈ h~r| ˆQb|~r 0i. (37)
In order to de ne the Bogoliubov transformation we will take advantage of two symmetry properties of the operator L, analogous to a single condensate case [51, 89]
5 TWO COMPONENT BOSE EINSTEIN CONDENSATE 24 where u1 = Ã σ1 0 0 σ1 ! , u3 = Ã σ3 0 0 σ3 ! , (39) and σ1 = Ã 0 1 1 0 ! , σ3 = Ã 1 0 0 −1 ! , (40)
are the Pauli matrices. Now suppose that all eigenvalues of the L operator are real. This is in fact an important assumption of dynamical stability [51] that should be veri ed for a given system. The symmetries (38) imply that if
|ΨR ni = |ua ni |va ni |ub ni |vb ni , (41)
is a right eigenvector of L to an eigenvalue En, then |ΨLni = u3|ΨRni is a left eigenvector to the
same eigenvalue En. Simultaneously u1|ΨR∗n i is a right eigenvector to an eigenvalue −En.
There are four eigenvectors of L corresponding to a zero eigenvalue, |φai 0 0 0 , 0 |φ∗ ai 0 0 , 0 0 |φbi 0 , 0 0 0 |φ∗ bi . (42)
The other eigenstates of L can be divided into two families "+" and "−", according to the sign of a norm de ned as
hΨR
n|u3|ΨRn0i = ±δn,n0. (43)
Having a complete set of eigenvectors of L we obtain an important completeness relation
ˆ1 = X n∈”+” |ua ni |va ni |ub ni |vb ni ³ huan|, −hvna|, hubn|, −hvnb|´+ X n∈”+” |va∗ n i |ua∗ n i |vb∗ ni |ub∗ ni ³ −hvna∗|, hua∗n |, −hvnb∗|, hub∗n|´ + |φa0ihφa0| 0 0 0 0 |φ∗ a0ihφ∗a0| 0 0 0 0 |φb0ihφb0| 0 0 0 0 |φ∗ b0ihφ∗b0| . (44)
The eigenvectors of the L operator de ne the Bogoliubov transformation ˆ Λa ˆ Λ† a ˆ Λb ˆ Λ†b = X n∈”+” ua n va n ub n vb n ˆcn+ X n∈”+” va∗ n ua∗ n vb∗ n ub∗ n ˆc † n, (45)
where quasi-particle excitations are described by operators that approximately ful ll the bosonic commutation relation [ˆcn, ˆc†n0] ≈ δn,n0. Employing the Bogoliubov transformation (45) we can diagonalize the eective Hamiltonian (33) to a simple form
ˆ Heff ≈
X
n∈”+”
Enˆc†nˆcn, (46)
where En are energies of the "+" family solutions to the Bogoliubov-de Gennes equations,
{ua
n, vna, ubn, vnb}. The quasi-particle excitation operators can be written as
ˆc†
n = huan|ˆΛ†ai − hvna|ˆΛai + hubn|ˆΛ†bi − hvnb|ˆΛbi, (47)
and their application, due to (36), does not change the particle number, contrary to the standard Bogoliubov theory.
5.2 Bogoliubov ground state in a particle representation
Ground state of the Hamiltonian up to the second order of the perturbative expansion (46) reveals no quasi-particle excitations
ˆcn|0Bi = 0, (48)
for all n ∈ ” + ”. Excited states can be generated acting with quasi-particle creation operators ˆc†
n on the Bogoliubov vacuum state |0Bi. The quasi-particle representation (46) is natural to
represent system eigenstates within the Bogoliubov theory. It is also suitable to nd low order correlation functions. The original particle representation is, however, much more convenient if we need predictions for density measurements. Simulations of atomic positions in a cloud can provide us with information about possible density uctuations.
The Bogoliubov ground state is a certain particle state so we assume that it can be obtained acting with some particle creation operators ˆd†
a and ˆd†b on the particle vacuum
|0Bi ∼ ³ ˆ d† a ´Ma³ ˆ d†b´Mb|0i. (49) If we require that ˆd†
a and ˆd†b commute with all quasi-particle annihilation operators [92],
5 TWO COMPONENT BOSE EINSTEIN CONDENSATE 26
then the Bogoliubov ground state is indeed annihilated by all quasi-particle annihilation opera-tors, ˆcn ³ ˆ d† a ´Ma³ ˆ d†b´Mb|0i =³dˆ† a ´Ma³ ˆ d†b´Mbˆcn|0i = 0. (51)
In the following we will show that the set of equations (50) is solved by particle creation operators of the form ˆ d† a = ˆa†0ˆa†0+ ∞ X α,β=1 Za αβˆa†αˆa†β, dˆ†b = ˆb†0ˆb†0+ ∞ X α,β=1 Zb αβˆb†αˆb†β, (52) where ˆa†
α (ˆb†α) are bosonic particle creation operators that create atoms in modes φaα (φbα)
orthogonal to the condensate wavefunction φa0 (φb0). Zαβa and Zαβb are symmetric matrices to
be found.
The operator ˆd†
a that accounts for creating pairs of atoms in the Bogoliubov vacuum was
introduced in a single condensate case in a variational approximation [96]. Later it was proved to produce an exact Bogoliubov vacuum both in homogeneous and inhomogeneous condensates [92, 97]. In a two component condensate the Bogoliubov state (49) was used in a system of two independent (i.e. noninteracting) condensates [98]. We will show that it can be applied also to a system with nonzero intercomponent interactions.
Substituting the ansatz (52) into (50) we obtain equations (see also (47)) hva n|φ∗aαi = ∞ X β=1 hua n|φaβiZβαa , hvbn|φ∗bαi = ∞ X β=1 hub n|φbβiZβαb , (53)
which, when multiplied by hφaγ|uani and hφbγ|ubni, respectively, and summed over n, are
trans-formed to hφaγ|ˆΓa|φ∗aαi = ∞ X β=1 hφaγ| ˆUa|φaβiZβαa hφbγ|ˆΓb|φ∗bαi = ∞ X β=1 hφbγ| ˆUb|φbβiZβαb , (54)
where ˆΓa, ˆΓb, ˆUa and ˆUb are built with quasiparticle modes
ˆ Γa = X n∈”+” |ua nihvna|, Γˆb = X n∈”+” |ub nihvbn|, ˆ Ua = X n∈”+” |uanihuan|, Uˆb = X n∈”+” |ubnihubn|. (55)
The completeness relation (44) implies that the ˆΓa and ˆΓb operators are symmetric and that
ˆ Ua= X n∈”+” |va∗ n ihvna∗| + ˆ1a⊥, Uˆb = X n∈”+” |vb∗ nihvb∗n| + ˆ1b⊥, (56)
where ˆ1a
⊥ and ˆ1b⊥are identity operators in subspaces orthogonal to condensate wavefunctions φa0
and φb0, respectively. We can write a reduced single particle density matrix for the component
a (and b analogously) h0B| ˆψa†(~r) ˆψa(~r0)|0Bi = Naφ∗a0(~r)φa0(~r0) + X n∈”+” va n(~r)va∗n (~r0). (57)
Comparing the rst of the equations (56) with (57) we can see that ˆUa is diagonal in a basis of
single particle density matrix eigenstates φaα. Similarly ˆUb in a basis built with φbα. Then one
immediately obtains solutions for the Za,b
αβ matrices Za αβ = hφaα|ˆΓa|φ∗aβi dNa α+ 1 , Zb αβ = hφbα|ˆΓb|φ∗bβi dNb α+ 1 , (58) where dNa,b
α are eigenvalues of the single particle density matrices, that is numbers of atoms
depleted from the condensate wavefunctions. For the component a we have h0B| ˆψa†(~r) ˆψa(~r0)|0Bi ≈ Naφ∗a0(~r)φa0(~r 0) + ∞ X α=1 dNa αφ∗aα(~r)φaα(~r 0). (59)
Note that the matrices Za
αβ, Zαβb , hφaα|ˆΓa|φaβ∗ iand hφbα|ˆΓb|φ∗bβiare symmetric. The ansatz (52) is
therefore self-consistent if the operators ˆΓa,bare also diagonal in a basis built with single particle
density matrix eigenvectors. We have proved analytically that this is the case in a homogeneous system, as well as for a two component BEC in a double well trapping potential. We have also con rmed that numerically for a system trapped in a spherically symmetric trap.
The nal form of the solution for the Bogoliubov vacuum state in the particle representation is |0Bi ∼ " ³ ˆa†0´2+ ∞ X α=1 λaα³ˆa†α´2 #Na/2 × " ³ ˆb† 0 ´2 + ∞ X α=1 λbα³ˆb†α´2 #Nb/2 |0i (60) where λa α = hφaα|ˆΓa|φ∗aαi dNa α + 1 , λb α = hφbα|ˆΓb|φ∗bαi dNb α+ 1 . (61)
6 Density uctuations close to a phase separation transition
6.1 Density measurement
A single particle density (i.e. the rst order correlation function) can give quite unexpected results if we ask about an atomic density measured in an experiment. In a numerical simulation
6 DENSITY FLUCTUATIONS CLOSE TO A PHASE SEPARATION TRANSITION 28
[99] it was shown that two condensates prepared in an initial Fock state acquire a relative phase during a density measurement process. A single experiment reveals interference fringes located at random positions, whereas the single particle density being an average over many measurements, remains at. This Fock state interference has been proved experimentally [100, 101] showing interference fringes between independent condensates. In a theoretical analysis the build-up of a relative phase was studied [102] and in an exact analytical work interference eects in second-and higher-order correlation functions were con rmed [103]. Situations when a single photo of a system may be signi cantly dierent from the averaged picture involve also collisions of two non-ideal condensates [104] a and condensate with a dark soliton [97, 105].
In order to perform a simulation of the density measurement we generally need the full many body probability density. As the number of particles grows, however, using that function to nd atomic positions quickly becomes a very formidable task. Instead one could use a sequential method from [99]. Having found successive atoms at certain positions one builds there a condi-tional probability density for nding an addicondi-tional one. Using this method within the Bogoliubov theory is however not so straightforward, since it would require inversion of the nonlinear trans-formation (47), unless we have written the Bogoliubov vacuum in the particle representation. We will however apply yet another method, which will clearly show us how density pro les are acquired as we perform the measurement on the Bogoliubov vacuum state. The approximate method was introduced for a single condensate [97] but since the two component ground state has a similar structure, we can use it also in the present case.
First, we can adjust phases of the eigenmodes of the single particle density matrices (see (59) and (61))
ϕaα(~r) = φaα(~r) e−iArg(λ
a
α)/2, ϕ
bα(~r) = φbα(~r) e−iArg(λ
b
α)/2, (62)
sa that the coe cients λa,b are real and positive. Then, we can restrict to only those modes that
have the largest values of the Bogoliubov vacuum coe cients (61) Λa,bα ≡ λ
a,b α
1 − λa,bα
À 1. (63)
The above adjustment of the phases makes it possible to rewrite the Bogoliubov vacuum state as a gaussian superposition over perfect condensate states [97]
|0bi ∼ Z dqadqbexp à − Ma X α=1 q2 aα 2Λa α ! exp − Mb X α=1 q2 bα 2Λb α |Na : φqai|Nb : φqbi, (64)
where |Na : φqai and |Nb : φqbi are many body states where, respectively, Na and Nb atoms
occupy single particle wavefunctions φqa(~r) = φ0a(~r) +√1Na PMa α=1qaαϕaα(~r) q 1 + √1 Na PMa α=1qaα , φqb(~r) = φ0b(~r) + √1Nb PMb α=1qbαϕbα(~r) q 1 + √1 Nb PMb α=1qbα . (65)
Consequently, results of a single measurement to a system in the state (60) can be approximated by the densities σa(~r) ∼ ¯ ¯ ¯ ¯ ¯φa0(~r) + 1 √ Na Ma X α=1 qaα ϕaα(~r) ¯ ¯ ¯ ¯ ¯ 2 , σb(~r) ∼ ¯ ¯ ¯ ¯ ¯ ¯φb0(~r) + 1 √ Nb Mb X α=1 qbα ϕbα(~r) ¯ ¯ ¯ ¯ ¯ ¯ 2 , (66)
where real parameters qaαand qbαhave to be chosen randomly, for each experimental realization,
from the Gaussian probability density P (qa, qb) ∼ Ma Y α=1 exp à −q 2 aα Λa α ! M b Y β=1 exp à −q 2 bβ Λb β ! . (67)
6.2 Density uctuations in a nite box
A two component homogeneous condensate is an example of a Bose system where the Bogoliubov theory gives analytical results even in the presence of a process which transfers atoms between the two components. For such a Josephson-like coupled system a linear stability analysis was performed to study the dispersion relation for the excitation spectra [106, 107], and a Bogoliubov transformation was derived with subsequent stability analysis with respect to uctuations of the relative number of atoms [108]. In the system preserving numbers of atoms in its components a dynamical instability leading to a phase separation was inferred from the quasi-particle excitation spectrum [106, 107, 109]. In the phase-segregated regime, the linearization of Gross-Pitaevskii equations revealed no quasi-particle excitations localized near the phase boundary [110].
In the following we will study the Bogoliubov vacuum state in the particle representation for a system approaching the phase separation transition. We will assume xed particle numbers in both components and repulsive interactions, i.e. ga, gb, g > 0. The condensates are con ned in
a box of L × L × L size with periodic boundary conditions. The ground state solution of the Gross-Pitaevskii equations (31) is φa0 = √1 L3, φb0 = 1 √ L3, (68)
and the chemical potentials
µa = gaρa+ gρb, µb = gbρb + gρa, (69)
where ρa,b = Na,b/L3 are densities of the a and b components. In a homogeneous system it is
6 DENSITY FLUCTUATIONS CLOSE TO A PHASE SEPARATION TRANSITION 30
equations of the form
ua k va k ub k vb k ei~k·~r √ L3. (70)
We obtain two quasi-particle excitation branches [109] Ek,± = ω 2 ak+ ω2bk 2 ± v u u t(ωak2 − ωbk2 ) 2 4 + ¯h2k4 mamb g2ρ aρb 1/2 , (71) where ω2 ak = ¯h2k2 2ma à ¯h2k2 2ma + 2gaρa ! , ω2 bk = ¯h2k2 2mb à ¯h2k2 2mb + 2gbρb ! . (72)
are the usual single condensate Bogoliubov dispersions. In the long wavelength limit k → 0 we have ωi,k ≈ ¯hcik where ci = √giρi is the sound velocity of the i = a, b condensate. The double
condensate dispersions are phonon-like in this limit,
Ek,± ≈ ¯hc±k, (73)
with sound velocities
c2±= 1 2 Ã c2a+ c2b ± s (c2 a− c2b) 2 + 4 g2 gagb c2 ac2b ! . (74)
The quasiparticle modes (solutions to the Bogoliubov-de Gennes equations) are given by ua k,± va k,± ub k,± vb k,± = 2gEkb(Eka+ Ek,±)√ρaρb 2gEkb(Eka− Ek,±)√ρaρb ³ E2 k,±− ω2ak ´ (Ekb+ Ek,±) ³ E2 k,±− ω2ak ´ (Ekb− Ek,±) χ±, (75) where Eka= ¯h2k2 2ma , Ekb = ¯h2k2 2mb , (76)
and the normalization factor χ± = ½ 4Ekb · 4EkaEkbg2ρaρb+ ³ E2 k,± − ωak2 ´2¸ Ek,± ¾−1/2 .
(77)
The reduced single particle density matrices (57) are diagonal in the ei~k·~r/√L3 basis. However,
in order to have the ˆΓa,b operators (55) also diagonal we have to switch to the basis
φa~ks= φb~ks = s 2 L3 sin ³ ~k · ~r´, φa~kc = φb~kc= s 2 L3 cos ³ ~k · ~r´. (78)
The Bogoliubov vacuum state in the particle representation reads then |0Bi ∼ ˆa† 0ˆa†0+ X ~k
λka³ˆa†~ksˆa~ks† + ˆa†~kcˆa~kc† ´ Na/2 × ˆb† 0ˆb†0+ X ~k λkb³ˆb~ks† ˆb~ks† + ˆb†~kcˆb~kc† ´ Nb/2 |0i, (79) where λa k= ua k,+vk,+a + uak,−vak,− ³ va k,+ ´2 +³va k,− ´2 + 1, λ b k = ub k,+vk,+b + ubk,−vk,−b ³ vb k,+ ´2 +³vb k,− ´2 + 1, (80)
and the operators ˆa†
~ks, ˆa†~kc ˆb~ks† and ˆb~kc† create atoms in the modes (78).
If the intercomponent interactions g are strong enough it is no longer energetically favorable to mix the condensates, the uniform solutions (68) to the Gross-Pitaevskii equations become unstable, and the two components spatially separate. In an in nite box (L → ∞ and Na,b→ ∞
but ρa,b= const) an imaginary eigenvalue in the Bogoliubov spectrum (71) appears for
g2 > gagb. (81)
In a nite box with periodic boundary conditions, however, the momentum of a quasiparticle can have only discrete values
~k = 2π
L (nx~ex+ ny~ey+ nz~ez) , (82)
where nx, ny, nz are non-zero integers. The minimal momentum value of a quasiparticle is 2π/L
and consequently the phase separation condition is modi ed g2 > Ã ¯h2π2 maL2 1 ρa + ga ! Ã ¯h2π2 mbL2 1 ρb + gb ! . (83)
This shows that for the nite system the minimal value of the interactions g leading to the phase separation has to be higher than the corresponding value for L → ∞. We will show that approaching the condition (83) one can observe density uctuations on a scale of the order of L. We consider a two component condensate of 87Rb atoms in two dierent internal states.
6 DENSITY FLUCTUATIONS CLOSE TO A PHASE SEPARATION TRANSITION 32 0 5 10 15 k L/(2π) 10-6 10-4 10-2 100 depletion 0 5 10 15 k L/(2π) -0.4 -0.3 -0.2 -0.1 0 λ (a) (c) 0 5 10 15 k L/(2π) 10-5 10-4 10-3 10-2 10-1 100 101 depletion 0 5 10 15 k L/(2π) -0.2 0 0.2 0.4 0.6 0.8 1 λ (b) (d)
Figure 9: Condensate depletion far (panel (a)) and close (panel (b)) to the phase separation transition point. Panels (c) and (d) show corresponding coe cients λa,b
k of the Bogoliubov
vacuum state (60). Red circles represent modes of the smaller condensate (Na = 5000) whereas
black circles of the larger one (Nb = 20000). The far-from phase separation scattering length is
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